Quasi-category theory you can use

Transcription

1 Quasi-category theory you can use Emily Riehl Harvard University eriehl Graduate Student Topology & Geometry Conference UT Austin Sunday, April 6th, 2014

2 Plan Part I. Introduction to quasi-categories. Quasi-categories are models for abstract homotopy theory Quasi-categories are good for homotopy limits and colimits Part II. Category theory of quasi-categories (developed by Joyal, Lurie, Nichols-Barrer, Gepner, Haugseng,..., R-Verity (v2.0). Universal properties in quasi-categories (case study: initial objects) General colimits in quasi-categories

3 Abstract homotopy theory Classical homotopy theory studies topological spaces up to (weak) homotopy equivalence. Abstract homotopy theory studies objects up to weak equivalence : given X Y think of X and Y as the same. E.g., chain complexes up to quasi-isomorphism spectra up to stable equivalence categories up to equivalences But it s hard to work in the homotopy category. Better to use: a Quillen model category, a quasi-category (aka -category).

4 Some recent work Precise statements of the following theorems are proven using quasi-categories. Theorem (Ben-Zvi & Nadler). S 1 -equivariant quasi-coherent sheaves on the loop space of a smooth scheme are de Rham modules. Theorem (Francis). Homology theories for topological n-manifolds are equivalent to n-disk algebras. Theorem (Barwick & Schommer-Pries). The homotopy theory of (, n)-categories is characterized up to equivalence by certain axioms, and its space of automorphisms is equivalent to (Z/2) n.

5 Quasi-categories A quasi-category is a simplicial set A with composition. A 0 A 1 A 2 A 3 Composition of 1-simplices: f g in A f g in A h

6 Examples of quasi-categories A topological space is a quasi-category: composites exist because any simplex deformation retracts onto each of its horns. A category is a quasi-category: 0-simplices are objects, 1-simplices are arrows, n-simplices are composable strings of n arrows. Some special quasi-categories: n = topological n-simplex = (0 1 n) S = ( ) = colim n S n Q: What homotopy theories do these present?

7 The homotopy category of a quasi-category The homotopy category ho(a) of a quasi-category A has: objects = vertices of A = elements of A 0 arrows = equivalence classes of 1-simplices f y y x f x g f g x y x g y x y x y x y g f g f E.g., the homotopy category of a space is its fundamental groupoid the homotopy category of a category is the category ho( n ) = (0 1 n) ho(s ) = ( )

8 Isomorphisms in a quasi-category An isomorphism in a quasi-category is a 1-simplex that represents an isomorphism in its homotopy category. Theorem (Joyal). Each isomorphism in a quasi-category A extends to a map S A. Data: x, y, x f y, y g x, f x y g g y x, x f y,...

9 Interlude: classical homotopy colimits X Y hocolimit cone Z P P = Z X X I X X I X Y hocolimit cone Z Y X W P P = Z I Z Y I Y X I X W I W... and initial with this data.

10 Initial objects, defn. A vertex a in a quasi-category A is initial iff any sphere in A with a as its starting vertex can be filled to a simplex. 0 a n n A a x x a y x a z x...

11 Initial objects, I & II defn I. A vertex a A is initial iff 0 a A is left adjoint to the constant functor A! 0. defn II. A vertex a A is initial iff a A A. (a A) 0 (a A) 1 (a A) 2 a x a x a y a a a x y z

12 Initial objects, III defn III. A vertex a A is initial iff there exists a map ɛ const a id A in A A. A const a A 1 ɛ A id A A a ɛx x x A 0 a x ɛ f f a y f A 1 a x a ɛ α y a z α A 2

13 Initial objects, IV defn IV. A vertex a A is initial iff 0 a A and A! 0 extend to a homotopy coherent adjunction. adjunction data: a A 0 and ɛ const a id A (A A ) 1 homotopy coherent adjunction data: higher simplices in A spanning the vertex a higher simplices in A A spanning the vertices const a and id A

14 Initial objects, V defn V. A vertex a A is initial iff for any f X A there exists a map const a f unique up to equivalence in A X Thus ɛ const a id A represents a unique equivalence class in A A.

15 Colimits, defn. A colimit of a diagram d A in a quasi-category A is an initial object in the quasi-category of cones under d. data: diagram x z y x y colimit cone z p The universal property of the colimit is encoded by the universal property of an initial object.

16 Colimits, I & II defn I. A quasi-category A has -shaped colimits iff the constant diagram functor A! A has a left adjoint A colim A. defn II. The map A colim A defines a -colimit functor in A iff colim A is equivalent to the quasi-category of cones.

17 Colimits, III defn III. A quasi-category A has -shaped colimits iff there exist maps ɛ colim! id A in A A and η id A! colim in (A ) A satisfying the triangle identities. colimit cone: A id A A 1 η A! colim A universal property: A colim! A 1 ɛ A id A A

18 Colimits, IV defn IV. A quasi-category A has -shaped colimits iff the constant diagram functor A! A and its left adjoint A colim A extend to a homotopy coherent adjunction. homotopy coherent adjunction data: higher simplices in each of the four hom-quasi-categories between A and A corresponding to strictly undulating squiggles:

19 Colimits, V defn V. A vertex p A and cone under d A define a colimit cone iff for any f X A and f-indexed cone under d, that cone factors uniquely (up to equivalence in A X ) through the colimit cone, inducing a map const p f. X! 0 f d A X! = A 0 d! f A! p! p A The universal property defines a unique 2-cell in the 2-category of quasi-categories.

20 References The 2-category theory of quasi-categories arxiv: Homotopy coherent adjunctions and the formal theory of monads arxiv: Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions arxiv: Categorical Homotopy Theory, Cambridge University Press, eriehl/cathtpy.pdf